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作者：Addario-berry, Louigi; Donderwinkel, Serte
作者单位：McGill University; University of Groningen; University of Groningen
摘要：We obtain new nonasymptotic tail bounds for the height of uniformly random trees with a given degree sequence, simply generated trees and conditioned Bienaym & eacute; trees (the family trees of branching processes) in the process settling three conjectures of Janson ( Probab. Surv. 9 (2012) 103-252) and answering several other questions from the literature. Moreover, we define a partial ordering on degree sequences and show that it induces a stochastic ordering on the heights of uniformly ran...
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作者：Minzer, Dor; Sah, Ashwin; Sawhney, Mehtaab
作者单位：Massachusetts Institute of Technology (MIT)
摘要：We prove that there exists a constant gamma (crit) approximate to 0.17566 such that if G similar to G (n, 1/2), then for any epsilon > 0 with high probability G has a equipartition such that each vertex has (gamma( crit) - epsilon) root n more neighbors in its own part than in the other part and with high probability no such partition exists for a separation of (gamma( crit) + epsilon) root n . The proof involves a number of tools ranging from isoperimetric results on vertex-transitive sets of...
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作者：Hansen, Benjamin; Mueller, Tobias
作者单位：University of Groningen
摘要：We consider percolation on the Voronoi tessellation generated by a homogeneous Poisson point process on the hyperbolic plane. We show that the critical probability for the existence of an infinite cluster is asymptotically equal to pi A/3 as A-* 0. This answers a question of Benjamini and Schramm ( J. Amer. Math. Soc. 14 (2001) 487-507).
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作者：Adhikari, Arka; Chatterjee, Sourav
作者单位：Stanford University; Stanford University
摘要：Consider a discrete one-dimensional random surface whose height at a point grows as a function of the heights at neighboring points, plus an independent random noise. Assuming that this function is equivariant under constant shifts, symmetric in its arguments, and at least six times continuously differentiable in a neighborhood of the origin, we show that, as the variance of the noise goes to zero, any such process converges to the Cole-Hopf solution of the 1D KPZ equation under a suitable sca...
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作者：Tourniaire, Julie
作者单位：University of Vienna
摘要：We consider a system of particles performing a one-dimensional dyadic branching Brownian motion with space-dependent branching rate, negative drift - mu and killed upon reaching 0, starting with N particles. More precisely, particles branch at rate rho /2 in the interval [ 0, 1], for some rho > 1, and at rate 1/2 in (1, +infinity ). The drift mu(rho) is chosen in such a way that, heuristically, the system is critical in some sense: the number of particles stays roughly constant before it event...
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作者：Bou-Rabee, Ahmed; Gwynne, Ewain
作者单位：New York University; University of Chicago
摘要：We prove a shape theorem for internal diffusion limited aggregation on mated-CRT maps, a family of random planar maps which approximate Liouville quantum gravity (LQG) surfaces. The limit is a LQG harmonic ball, which we constructed in a companion paper. We also prove an analogous result for the divisible sandpile.
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作者：Andres, Sebastian; Gantert, Nina; Schmid, Dominik; Sous, Perla
作者单位：Braunschweig University of Technology; Technical University of Munich; University of Bonn; University of Cambridge
摘要：We study biased random walks on dynamical percolation on Zd. We establish a law of large numbers and an invariance principle for the random walk using regeneration times. Moreover, we verify that the Einstein relation holds, and we investigate the speed of the walk as a function of the bias. While for d = 1 the speed is increasing, we show that, in general, this fails in dimension d >= 2. As our main result, we establish two regimes of parameters, separated by an explicit critical curve such t...
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作者：Van Engelenburg, Diederik; Hutchcroft, Tom
作者单位：University of Vienna; California Institute of Technology
摘要：We prove that if a unimodular random rooted graph is recurrent, the number of ends of its uniform spanning tree is almost surely equal to the number of ends of the graph. Together with previous results in the transient case, this completely resolves the problem of the number of ends of wired uniform spanning forest components in unimodular random rooted graphs and confirms a conjecture of Aldous and Lyons (2006).
